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323 Collisions between elastic bodies: Newton's cradle D R Lovett, K M Moulding and S Anketell-Jones Department of Physics. University of Essex. Wivenhoe Park. Colchester CO1 3SQ. UK Received 22 February 1988 Abstract A form of Newton's cradle is described in which spheres of different sizes can he used and overall collision times measured using a BBC microcomputer. It i\ shown that collision times cannot be explained on the basis of collisions between initial11 non-touching pairs of 9pheres. but that Hertz's theory is applicable. As the collision perturbation passes down the chain. the additional collision times for subsequent collisions are half the collision time for the first collision. 1. Introduction Newton's cradle (see figure l), which consists of five metal balls suspended by threads from a metal frame such that all five balls just touch when hanging stationary. is well known and is widely available as a commercial toy. It also forms the basis of an experi- ment in the materials and mechanics unit of the Revised Nuffield Advanced Science Course (Harris 1985). If one of the end balls is displaced and then allowed to collide with the chain of remaining balls. the motion in principle can be described by the laus of conservation of energy and momentum. The usual way to consider the problem is to assume ;L long ball chain with a small separation (F) between each ball (Piquette 1984). Any perturbation to the chain produced at one end is propagated without change of shape along the chain; i.e. the behaviour of the chain is free from dispersion. If the balls are touching, then in order to solve for the subsequent motion one must assume that there is no dispersion (Hermann and Schmalzle 1981). It has been shown by computer simulation (Herrmann and Seitz 1982) that in the case of touching elastic balls the first set of collisions is nearly but not totally dispersion-free: however. after this the balls remain moving and slightly separated such that all subsequent collisions are dispersion-free. We have repeated the simula- tlon. and report here an undergraduate project set Zusammenfassung Eine Art 'Newton's crndle' (Kugelstossgerit) wird beschrieben. hei der Kugeln unterschiedlicher Grcjw vcrwendet unci Gesamtstosszeiten gemessen werden kOnncn untcr Verwendung cine\ BBC-hli~rocomputers. Es \vird gezeigt. dass die Stosszeiten nicht auf Grund von Stci\\cn zuischen ursprunglich sich nicht lwuhrenden Kugelpaaren erklirhar sind. dnss jedoch dic Hertzsche Theorie anwendbar ist. Die zuslitzlichen Stosszciten fur alle nachfolgendcn Stossc Innerhalb der Stossfolge sind nur halh \o ;roar \vie die Stosszeit flir den crsten Stohr. up to carry out further investigation of collisions between the elastic balls in a Newton's cradle. 2. Experimental arrangement A cradle was built in which balls of different sizes couldbeinterchanged. It wasmadewith a heavy base to ensure stability. Small rings were attached to the balls and the balls were suspended by nylon fishing line. which was hung over threaded rods (studding). These rods had a flat surface milled on them so that the balls could suing without the nylon line catching on the threading. The use of threaded rod enabled easy and accurate lateral adjustment of the positions of the balls. Vertical adjustment of the balls was achie\,ed by the use of small hooked steel rods (rather like 'Allen' keys) connected to nuts which could rotate on the threaded rods. Each length of line looped over one of these steel bars the height of which could be accurately altered by rotation of the corresponding nut. In this aay. the height of each ball could be adjusted individually. The arrangement is illustrated in figure 2. Thus balls of any size could be lined up on the stand such that they just touched and their centres were at the same height. Collision times were mea- sured electrically by attaching thin wires to the balls.

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323

Collisions between elastic bodies: Newton's cradle

D R Lovett, K M Moulding and S Anketell-Jones

Department of Physics. University of Essex. Wivenhoe Park. Colchester CO1 3SQ. UK

Received 22 February 1988

Abstract A form of Newton's cradle is described i n which spheres of different sizes can he used and overall collision times measured using a BBC microcomputer. I t i\ shown that collision times cannot be explained on the basis o f collisions between initial11 non-touching pairs of 9pheres. but that Hertz's theory is applicable. As the collision perturbation passes down the chain. the additional collision times for subsequent collisions are half the collision time for the first collision.

1. Introduction Newton's cradle (see figure l ) , which consists of five metal balls suspended by threads from a metal frame such that all five balls just touch when hanging stationary. is well known and is widely available as a commercial toy. It also forms the basis of an experi- ment in the materials and mechanics unit of the Revised Nuffield Advanced Science Course (Harris 1985). I f one of the end balls is displaced and then allowed to collide with the chain of remaining balls. the motion in principle can be described by the laus o f conservation of energy and momentum. The usual way to consider the problem is to assume ;L

long ball chain with a small separation ( F ) between each ball (Piquette 1984). Any perturbation to the chain produced at one end is propagated without change of shape along the chain; i .e. the behaviour o f the chain is free from dispersion. If the balls are touching, then in order to solve for the subsequent motion one must assume that there is no dispersion (Hermann and Schmalzle 1981). It has been shown by computer simulation (Herrmann and Seitz 1982) that in the case of touching elastic balls the first set of collisions is nearly but not totally dispersion-free: however. after this the balls remain moving and slightly separated such that all subsequent collisions are dispersion-free. We have repeated the simula- tlon. and report here an undergraduate project set

Zusammenfassung Eine Art 'Newton's crndle' (Kugelstossgerit) wird beschrieben. hei der Kugeln unterschiedlicher Grcjw vcrwendet unci Gesamtstosszeiten gemessen werden kOnncn untcr Verwendung cine\ BBC-hli~rocomputers. Es \vird gezeigt. dass die Stosszeiten nicht auf Grund v o n Stci\\cn zuischen ursprunglich sich nicht lwuhrenden Kugelpaaren erklirhar sind. dnss jedoch dic Hertzsche Theorie anwendbar ist. Die zuslitzlichen Stosszciten f u r alle nachfolgendcn Stossc Innerhalb der Stossfolge sind nur halh \o ;roar \vie die Stosszeit flir den crsten Stohr.

up to carry out further investigation of collisions between the elastic balls in a Newton's cradle.

2. Experimental arrangement A cradle was built in which balls of different sizes could be interchanged. I t was made with a heavy base to ensure stability. Small rings were attached t o the balls and the balls were suspended by nylon fishing line. which was hung over threaded rods (studding). These rods had a flat surface milled on them so that the balls could suing without the nylon line catching on the threading. The use of threaded rod enabled easy and accurate lateral adjustment o f the positions of the balls. Vertical adjustment of the balls was achie\,ed by the use of small hooked steel rods (rather like 'Allen' keys) connected to nuts which could rotate o n the threaded rods. Each length of line looped over one of these steel bars the height of which could be accurately altered by rotation of the corresponding nut. I n this a a y . the height of each ball could be adjusted individually. The arrangement is illustrated in figure 2 .

Thus balls of any size could be lined up on the stand such that they just touched and their centres were at the same height. Collision times were mea- sured electrically by attaching thin wires to the balls.

Page 2: Collisions between elastic bodies: Newton's cradle

324 D R Lovett et a1

W

Figure 1 Newton’s cradle.

The leads were connected to input ports of a BBC model B microcomputer in order to use the micro- second timer within the computer. The usual arrangement was to connect the two end balls to separate input lines and to connect the intermediate balls to earth, although for two-ball interactions a single input line was used. Ball-bearing of diameters 3.81 cm and 2.54 cm were used. (Ball-bearings of smaller size were tried but these had insufficient mass to give adequate stability; in particular, it was difficult using smaller balls to establish electrical contact between touching balls.)

3. Linear chains of two, three, four and five The time intervals were measured between the instant when a first ball touched the chain and the instant a last ball left the chain. This time interval

Figure 2 Experimental arrangement of the cradle.

will be referred to as the total interaction time. The velocity of impact of the first ball was varied by releasing it from different heights. The thin release cord was calibrated with displacement markers and the velocity of impact calibrated carefully from the height of release. The measured time intervals (the total interaction times) became shorter as the velo- city of impact increased, but for a fixed velocity the interaction time increased linearly with the number of balls in the chain. Figure 3 shows the results for balls of diameter 2.54 cm.

The program allowed for the very small delay which arose from the timing and computation (less than lops). With this step included, each graph passed accurately through the origin. For the case of two balls only, the interaction time is not zero as the balls remain touching for a finite time. An important feature to note from the graphs, therefore, is that the disturbance propagates through the chain at a velocity which is less than one tenth the speed of sound through the same length of steel. Assuming a velocity of 5850 m s-l for the steel balls (for this article, data for stainless steel have been taken from Kaye and Laby 1986), the time for sound to pass through five balls would be about 21ps. The meas- ured interaction time for a chain consisting of five balls varied from 190ps for an impact velocity of 1.1 m S” to 3 0 0 ~ s for an impact velocity of 0.07ms”. Hence the time for the end ball to separate is not given simply by the time for sound waves to propagate through the balls in opposite directions, reflect off the end surfaces, and return to meet at the point of separation of the balls. However, it is important to note the linearity of the graphs and the consequent fact that the velocity is transmitted uniformly down the chain even though this is much slower than the velocity of sound in steel.

The explanation lies in the application of Hertz’s law (Hertz 1881, but see Landau and Lifshitz 1959,

Reference polnt for dlsplacernent of first ball

Page 3: Collisions between elastic bodies: Newton's cradle

Collisions between elastic bodies: Newton's cradle

- 0.3 L

values (m S"]

7 0 . 0 7

0 1 2 3 4 5 Length o f chaln (number of balls)

Figure 3 Total interaction times for chains of two. three, four and five balls. The diameter of each ball = 0.0254 m.

Leroy 1985). When two balls collide, there is defor- mation of each ball. They come into contact over a small but finite portion of their surfaces. The area of contact increases until nearest approach of the centres occurs. Hertz's law arises from taking into account the distribution of pressure over this region of contact. The change of separation of the centres of the spheres ( h ) is obtained as a function of the total force ( F ) between the balls. It was mathernati- cally derived by Hertz that h is proportional to FY3. Integrating from h = 0 to h = h,,, where h,, corres- ponds to the change of separation of the centres of the spheres for maximum approach, allows calcula- tion of the, time 7 during which the collision takes place. Hertz's law takes the form

r = 2.94"~'ik'u)' '~ (1)

where U is the velocity of impact of the incoming spherical ball. ,U is the reduced mass, i.e. ,U = rnM/(m + M ) where m and M are the masses of the two colliding spheres. k=4Rf1' ' /5D, where R' is the reduced radius = rR / ( r+ R ) and r and R are the individual radii of the spheres. D is given by

D = (1 - a')/2E where U is Poisson's ratio and E is Young's modulus (the elastic constant) for the spheres, which are assumed here to be made from the same material.

325

For spheres of the same radii, Hertz's equation reduces to

t = 3.29(1- $)"5(M'/RE'~)"5 (2)

where M and R are the mass and radius respectively of each ball. The time of propagation of the motion along the chain of balls arises from the time taken to compress the spheres, the build-up of elastic waves, and the eventual transfer of energy to the next sphere.

To test the applicability of Hertz's law. we plot the interaction time T against (velocity of collision)-"' as given in table 1 for collisions between two balls; see figure 4. T = t in this case. The full line gives the theoretical relationship, tak- ing the density of stainless steel to be 7800 kg m-' (this value gives a correct mass of 67 g for each steel ball), Young's modulus as 215 GPa and Poisson's ratio as 0.28(3). It can be seen that there is good agreement between the experimental data and the theoretical relationship. There appears to be a small systematic deviation at lower velocities. although this deviation is within experimental error. Such deviation might be expected at low velocities where small frictional and other energy losses will be pro- portionally higher; however, a quantitative treat- ment of this was not undertaken. Note that a value of 1.70 (U)"" corresponds to a velocity of 0.07 m S" and a displacement of the impacting sphere of only 1 cm.

Plotting data for chains of three, four and five balls (where the numbers include the incoming ball) also gives linear relationships between the total interaction time and (velocity of collision)"". However, the ratios of the total interaction times for chains of two balls, three balls, four balls and five balls are 1:1.5:2:2.5 and not 1:2:3:4 as would be expected if the interaction times arose from a series of independent collision times added together. The addition of any further ball onto the chain adds a time interval to the interaction time of half the interaction time ( r ) for a two-ball collision. This shows that the interaction times for each pair of neighbouring balls overlap. Therefore, the chain cannot be broken down into components as if each

Table 1 Variation of interaction times with impact velocity and (impact velocity)"" for two-ball collisions.

Velocity of impact. Interaction time. L) (m S") 5 (ms) ( U ) ' ( 0.07 (l. 113 1.70 0.14 0.106 1.48 0.29 0.097 0.43

1 . 2 0.091 I . 18

0.59 0.084 1 . 1 1 0.75 0.083 0.91 0.078

1.06

1.10 0.077 1.02 0.98

Page 4: Collisions between elastic bodies: Newton's cradle

326 D R Looett rt ul

- / / / /

0 1 2 ( V , m p r : + ) - " j

Figure 4 Total interaction time \)ersus (impact velocity) ' to test the applicability of Hertz's law. The impact velocity is in units of m S ~ I .

ball is separated by a small distance E . This conclu- sion has been stated already by Herrmann and Schmalzle (1984).

4. Collisions involving a non-uniform chain Chains of up to five balls were investigated where either the first or the last ball was of diameter 3.81 cm and the other balls were of diameter 2.54 cm as used before. Collision experiments were repeated

using the same procedure as previously. Some typi- cal data for the total interaction times are shown in table 2 and are plotted against chain length in figure 5 . A5 we would expect, the interaction time for a small ball colliding with a larger ball is the same as that for a larger ball colliding with a small ball. This is not the case once there are further small balls in the chain. We now apply Hertz's equation in the form of equation (1) for colliding spheres of differ- ent masses and radii. In the case of the heavier ball

Table2 Variation of interaction times with chain length for a nun-uniform chain. A large ball ( L ) is placed either last or first in the chain and the remaining balls (5 ) are small.

Interaction time. r (ma)

Velocity 2 balls 3 balls 4 balls 5 balls of impact I > (m s l ) S-L L-S \-\-L L-a-a S-5-S-L L-5-a-a a-S-s-s-L L-S-S-.;-S

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Collisions between elastic bodies: Newton's cradle 327

V e l o c l t y o f l n c o m l n g ball I m s"1 V e l o c l t y

o f lncomlng bal l 1ms"l

(a1 l l b l I I I I I ~ I I I 1 1 ,

0 0.10 0.16 0 0.10 0.16 Length of chaln l m ) Length o f c h a l n ( m )

making the initial impact, the velocities after this first collision. assuming a two-ball interaction, would be increased by the ratio 2 M / ( r n + M ) . The larger mass of the incoming ball leads to an increase in velocity when the momentum is exchanged with the small mass, even though in this case the larger sphere rebounds with significant speed. In this experiment. the larger sphere had mass 226 g. This results in a velocity ratio of 1.55 assuming a straight- forward elastic collision between the two spheres and that at this stage in the collision process no significant amount of momentum has been exchanged with subsequent balls. This should reduce the subsequent collision times by a factor of (1.55)"5 or 1.09. In figure 5 ( a ) , the heavy sphere is placed last in the row and the velocity correction is no longer relevant. In figure 5(b) . the heavy sphere is the impacting sphere, and the increased velocity applies after the first collision. The increase in gradi- ent and hence the increase in interaction time between case ( a ) and case ( h ) is approximately 1.1 and agreement between theory and experiment is within experimental error. Note that although the graphs extrapolate back close to the origin. the linear arguments and the comparison apply from the first collision onwards. If a different-sized ball is inserted inside the chain of otherwise equi-sized balls. linearity is maintained either side but broken where the interaction between unequal balls takes place.

Figure5 Total interactionfime for a mixed chain when (a) the last ball is larger. ( h ) the impacting ball is larger

5. Further discussion The results indicate that Hertz's law applies closely for the interaction of the spheres in Newton's cradle. This is so whether equi-sized balls are used or balls of different masses. In the latter case a simple assumption of exchange of energy and momentum on the basis of an elastic two-body collision obtains a suitable value of velocity for the use of Hertz's law in subsequent collisions. A more detailed analysis would necessitate dropping the assumption of an elastic collision.

It is perhaps surprising that the additional colli- sion times for collisions subsequent to the first collision (i.e. as one adds further spheres of the same mass) are equal to half the collision time for the first collision. This ratio is exact within the experimental errors of the present experiment. However, from the derivation of Hertz's theory (Leroy 1985). the collision time during which two balls remain in contact is just twice the time needed for the distance of approach of the centres ( h ) to reach a maximum. It is reasonable to suppose that the time required for the interaction to be passed on down the chain is directly related to the time required to achieve this distance of maximum h and minimum separation of the centres of the two balls.

The form of Newton's cradle discussed here in which spheres of different sizes can be interchanged provides an interesting set-up for the investigation of collisions between bodies and is suitable for

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328 D R Lovett et a1

student project work. Hertz’s law is applicable also to cylinders and these can be used to replace spheres.

References Harris J (ed.) 1985a Revised Advanced Nuffield Physics,

- 1985b Revised Advanced Nuffield Physics, Teachers Students Guide 1 (London: Longrnan) p 58

Guide I (London: Longrnan) pp 6-67

Herrmann H and Schmalzle P 1981 A m . J . Phys. 49 761 - 1984 Am. J . Phys. 52 84 Herrmann H and Seltz M 1982 Am. J . Phys. 50 977 Hertz H 1881 J . Reine Angew. Math. 92 156 Kaye G W C and Laby T H 1986 Tables of Physical

Constant 15th edn (London: Longman) Landau L D and Lifshitz E M 1959 Theory of Elasticity.

Course of Theoretical Physics v01 7, Transl. J B Sykes and W H Reid (London: Pergamon) p 30

Leroy B 1985 A m . J . Phys. 53 346 Piquette J C and Wu M-S 1984 A m . J . Phys. 52 83